NBPTS MC Math cheat sheet

Number and operations: a fraction a/b is a rational number when a and b are integers and b ≠ 0; division by zero is undefined, so any problem using it must be rejected as ill-posed.

Proportional reasoning rule: two quantities are proportional only when their ratio is constant; a table is proportional iff every y/x pair is equal and the graph passes through the origin.

Function definition: a relation is a function iff each input in the domain maps to exactly one output; vertical-line test fails precisely when one input has two outputs.

Geometry: sum of interior angles of an n-gon is (n − 2)·180°; the exterior-angle sum is 360° for every convex polygon regardless of n.

Statistics decision rule: mean is pulled by outliers, median is resistant; report median with interquartile range for skewed data, mean with standard deviation for roughly symmetric data.

Probability: for independent events P(A and B) = P(A)·P(B); for mutually exclusive events P(A or B) = P(A) + P(B), and P(A and B) = 0.

Place-value error diagnosis: a student who writes 3.5 as larger than 3.45 is comparing digit strings, not quantities; remediation targets magnitude and number-line placement.

Pedagogical content knowledge includes anticipating student strategies (counting on, making ten, partial sums) and choosing tasks that surface the underlying structure, not just correct answers.

Each ePortfolio entry is scored independently against its own rubric; a strong analysis in one entry cannot compensate for a missing required artifact in another.

Analysis must cite specific, dated evidence from the submitted work samples - names of students replaced with identifiers, direct quotations of student reasoning included.

A claim of impact requires a before-and-after comparison: baseline evidence, the instructional decision, and post-instruction evidence tied to the same learning goal.

Reflection must explain why the chosen instructional move was appropriate for those learners, including what was considered and rejected, not merely what was done.

Video entries are judged on evidence of student mathematical discourse and the teacher's questioning moves; editing that hides the teacher's decisions weakens the entry.

Accomplished-practice evidence must show differentiation for specific learners (English learners, students with disabilities, advanced learners) with the reasoning behind each adaptation.

Word-limit compliance is part of the rubric: exceeding a stated maximum can result in the response being unscorable, so plan the structure before drafting.

Assessment center exercises are timed, computer-delivered prompts written to the certificate area; each is scored holistically on a rubric scale, typically 0–4 or 1–4.

Rubric levels are defined by quality descriptors: a top score shows accurate mathematics, explicit reasoning, and attention to student thinking; a mid score is accurate but thin.

A zero is reserved for responses that are off-topic, illegible, or that fail to address the prompt at all - accuracy alone does not guarantee a passing level.

Prompts often require both solving a mathematics task and explaining the pedagogy; omitting either half caps the score regardless of the other half's quality.

Showing work matters: a correct answer with no reasoning can score lower than a correct answer with a clear, generalizable justification.

Time allocation rule: skim all prompts, budget minutes per prompt proportional to the number of required parts, and reserve time to check arithmetic and units.

When a prompt asks for an instructional decision, name the misconception, the evidence that reveals it, and the specific task or question that would address it.

Proposition 1: teachers are committed to students and their learning - evidence includes high expectations for every learner and knowledge of how students develop mathematically.

Proposition 2: teachers know the subjects they teach and how to teach them - evidence includes representing a concept multiple ways and anticipating misconceptions.

Proposition 3: teachers manage and monitor student learning - evidence includes varied assessment, clear learning goals, and adjusting instruction from evidence.

Proposition 4: teachers think systematically about their practice and learn from experience - evidence includes reflection, professional reading, and revision of a lesson after data.

Proposition 5: teachers are members of learning communities - evidence includes collaboration with colleagues and families and use of community resources.

Alignment rule: every portfolio claim should be traceable to a named Core Proposition or a specific mathematics standard, and the artifact should be the proof, not the assertion.

Mathematics standards emphasize process goals alongside content: problem solving, reasoning and proof, communication, connections, and representation must appear in the evidence.

Developmental progression in number: counting and cardinality, then additive reasoning, then multiplicative reasoning, then proportional and algebraic reasoning; instruction that skips a stage produces fragile procedures.

Concrete–representational–abstract sequencing is a decision rule: introduce with manipulatives, move to drawings and number lines, then to symbols, and check that the student can translate among all three.

Early-childhood learners need one-to-one correspondence and conservation of number before formal addition; middle-grade learners need integer and rational-number sense before solving equations.

Adolescent learners develop proportional and functional reasoning; tasks should require them to distinguish additive from multiplicative comparisons explicitly.

English learners need language supports tied to mathematics: sentence frames for justifying, visual glossaries, and paired discussion before written explanation.

Students with disabilities benefit from explicit strategy instruction, worked examples, and concrete supports; accommodation decisions should preserve the mathematical demand of the task.

Formative check rule: if fewer than roughly three-quarters of students can explain the reasoning behind a procedure, reteach the concept rather than assigning more practice.

Evidence hierarchy: student work and video showing actual discourse outrank teacher description; a claim without an artifact is treated as an assertion, not evidence.

Anonymity rule: remove student names, school names, and identifying details from every artifact; identifiable material can render an entry unscorable.

Each entry should answer three questions in order: what was the learning goal, what did students actually do, and what does that reveal about their understanding.

Rubric language is the drafting target: extract the descriptor verbs (analyze, justify, connect) from the rubric and make sure the response performs each one explicitly.

Triangulation strengthens a score: pair a written artifact with a video excerpt or assessment data so the same conclusion is supported by more than one source.

Feedback loop evidence: show the feedback given, the student's revision, and the change in understanding - a graded paper alone does not demonstrate impact.

Consistency check before submission: every artifact cited in the analysis must be present in the entry, and every artifact present must be analyzed somewhere.